|Q3. Expected value, variance and the mgf of the sample mean Let X1,..., X, be independent and identically distributed random variables with mean E(X;) = µ and the variance Var(X;) = o?, i = 1,..., n. 1. Let X = !E-1 X;. Use the properties of the expectation and the variance to show that E(X) = µ and Var(X): 2. Suppose X1,..., X, N(H,0²). Then the mgf of X; is given as Мx. (() — ех ut + i = 1,...,n.
|Q3. Expected value, variance and the mgf of the sample mean Let X1,..., X, be independent and identically distributed random variables with mean E(X;) = µ and the variance Var(X;) = o?, i = 1,..., n. 1. Let X = !E-1 X;. Use the properties of the expectation and the variance to show that E(X) = µ and Var(X): 2. Suppose X1,..., X, N(H,0²). Then the mgf of X; is given as Мx. (() — ех ut + i = 1,...,n.
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Transcribed Image Text:|Q3. Expected value, variance and the mgf of the sample mean
Let X1,..., X, be independent and identically distributed random variables with mean E(X;) = µ and the
variance Var(X;) = o?, i = 1,..., n.
1. Let X = !E-1 X;. Use the properties of the expectation and the variance to show that
E(X) = µ
and
Var(X):
2. Suppose X1,..., X, N(H,0²). Then the mgf of X; is given as
Мx. (() — ех
ut +
i = 1,...,n.
Use the independence of the random variables Xị, i = 1,..., n, to show that the mgf of X is
Mx(t) = exp ( ut +
2 n
From the mgf of X, identify the distribution of X.
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